I was running into a situation where my optimization (minimization) contain 2 integer variable and some real variables.
If we already know the number of combination the 2 integer variables can form (for example 100 cases) does this mean the worst case complexity (Big O notation) would involve a complete minimization of all 100 sub optimization problems ?
Thank you !
First, I think you may be misunderstanding Big O complexity. It's largely an asymptotic bound on computational effort as a function of problem size where problem size is assumed to be able to grow. So if you are saying that the number of combinations of the integer variables would be fixed at 100 while the number of real variables could/would increase, a Big O bound might be appropriate. Otherwise, I'm not sure what you mean.
In any case, given 100 combinations of values for the integer variables, you would not necessarily need to optimize all 100 subproblems. It might be obvious at the outset that some of the 100 combinations lead to infeasible solutions, while for other combinations the effect of the integer variables on the objective function might be so severe that there is no hope that any solution to the continuous variables would result in an optimal solution. This is problem-specific.